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Today I thought I’d share a video that I came across the other day. Hope you enjoy!

“Bacteria and viruses hitch a ride inside droplets of all kinds—sneezes, raindrops, toilet splatter. By reviewing footage of different types of drops, applied mathematician Lydia Bourouiba records and measures where they disperse in order to better understand how diseases spread. Watch how Bourouiba designs tests—some inescapably humorous and awkward—to study infectious disease transmission.”

Viruses are effective carriers of information. This information is called RNA (sort of like single stranded DNA), which is held in a vessel called the capsid. As it is favourable for the virus to be as efficient as possible, the capsid is made up of a repeating structure of proteins, which is coded by the RNA which it holds. Due to this repeating structure, the capsid has rotational symmetry.

These constraints on the organisation of the capsid proteins mean that they can only have helical or spherical structures. Thus, there are only a finite number of symmetries that they can have in 3-dimensions, which are embodied by the Platonic Solids.

Most viruses exhibit icosahedral symmetry, meaning they have the same properties as the icosahedron:

formed of 20 triangles

60 rotational symmetries

Source: Wikipedia

Another example is the common cold and HPV (Human papillomavirus), which are both caused by viruses of different sizes, but have the same symmetry: 6-fold rotational symmetry.

HPV | Source: Wikipedia

However, a more interesting group of viruses are those with 5-fold symmetry, as the structure of these is much harder to determine.

Structure via Symmetry

Symmetry is the key to finding the structure of a viruses, and as they are too small to see in microscopes, their structure can be determined using a technique called Cryo-electon microscopy, which is a bit like unfolding the shape and laying it out flat.

For structures with a 5-fold symmetry axis, this technique is almost impossible to use. An analogy of this is “trying to tile a bathroom floor: it is possible with hexagons but not with pentagons.” This mathematical phenomenon means that it is therefore necessary to know the overall symmetry of structures with 5-fold symmetries before their structure can be determined using cryo-EM.

But what is the reason for all this symmetry?

“Symmetry goes hand in hand with stability. Symmetric configurations are usually more stable, a feature that is important for viruses when transporting their genomic material between hosts.”

Today’s post will be part 2 of the series about the application of mathematics to the eight great technologies, according to the UK government. Click here to read part 1 of this series.

Regenerative medicine

Regenerative medicine studies the molecular and cellular processes that control the development of new, healthy tissue. This field aims to treat and cure diseases by discovering the mechanisms used by nature to restore the structure and function of damaged tissues.

Mathematical modelling can provide “qualitative insight” on how these mechanisms work, as well as help in the development of new flexible materials. Mathematics has other applications to medicine, for example medical statistics and in problems on medical imaging.

Agricultural science

Agricultural science is a key part of the world’s food supply system as it researches how we can sustainably, profitably and ethically produce food worldwide. Recently the UN has said that “if the population continues to rise at its present rate, then the world food output must increase by 70% by 2050”, hence agricultural science is more relevant and important than ever.

Thermodynamic and fluid mechanics is essential to understanding the fundamental processes of growing, freezing, cold storing, cooking, freezing and digesting food. Furthermore, the issue of feeding the worlds growing population requires the mathematics of optimisation, an area in computer science and operational research.

Advanced materials and nanotechnology

With modern technology, meta-materials – materialsthat have a myriad of mechanical, electrical, thermal and other properties – can be manufactured. These are materials that cannot be found naturally in nature.

An example of a meta-material is a material with a negative refractive index, which can therefore cause backward propagation of light.

Source: nature.com

The mathematics involved in designing such materials is quite complex and challenging, and can be used to analyse materials as ancient as rocks to modern day carbon fibres.

Energy and its storage

Mathematical understanding of stochastic processes is essential in communications science. This is due to the fact that the large number of users gives rise to random patterns of calls, emails, etc., which a network has to be able to deal with. The mathematics developed for communication networks can be applied to energy systems, as explained by Stan Zachary:

“If we integrate renewable energies, such as wind power, in the electricity grid, there will also be uncertainty, as we don’t know what the wind will be doing tomorrow. This will make planning and scheduling much more challenging and it will take sophisticated mathematics to get it right.”

Furthermore electricity must be consumed as soon as it is purchased, as it cannot be stored in large quantities, which presents a further challenge.

These challenges, which mathematicians can combat, are going to increase in the future as we shift towards low-carbon energy supplies, a more distributed supply network, electric vehicles, and the SMART Grid (where users have greater control over their energy demands and in turn supply more information to the grid company).

In 2012, the UK government published a list of eight Technologies in which the UK is “set to be a global leader”. In this 2 part series I will detail how maths is essential for ALL these great technologies.

The eight great technologies are:

big data and energy-efficient computing

Satellites and commercial applications of space

robotics and autonomous systems

synthetic biology

regenerative medicine

agri-science

advanced materials and nanotechnology

energy and its storage

Big Data

Big Data is a term used for data sets that are so large or complex that traditionally processing applications are inadequate.

In the UK, examples of where Big Data arises include data on prescription drugs (connecting origin, location and time of each prescription) and well as joining up data in order to allow authorities to recognise certain patterns and therefore improve public services accordingly.

There is a large challenge in visualising, modelling and understanding Big Data. How do we experiment on the systems that generate it and how do we control these systems? The mathematical challenges behind these questions require automation, which in turn relies on mathematical algorithms.

Mathematical techniques to deal with big data are being developed and researched, including network theory. Network theory describes nodes, that are linked together by edges. When dealing with large networks, it is hard to identify clusters – groups of highly interlinked nodes – or to segment the data into groups that share common features. However, network theory provides algorithms for both these problems.

Furthermore, more obscure areas of maths can aid in the analysis of Big Data, which can also take the form of images:

Algebraic topology is concerned with studying shapes using algebra and plays are very useful role in classifying images;

Category Theory, which investigates mathematical structures and concepts on an abstract level, can be used to split up an image in order to analyse it and see how the various components fit together.

This can aid machines ‘perceive’ what the images are and hence make decisions about it.

Satellites and Space

There are many areas in which mathematics can aid in space exploration, including:

Understanding and controlling the dynamics of satellite systems in order to efficiently place orbits.

Robotics and Autonomous Systems

Numerical methods developed by mathematicians are used to stimulate movement and control robotic systems. Furthermore, mathematics can be applied the field of robotics through machine learning algorithms, pattern recognition techniques, neural networks, which mimic simple nervous systems, and computer vision.

Genomics and Synthetic Biology

Genomics is a field in biology in which DNA sequencing methods and bioinformatics are used to sequence, assemble, and analyse the function and structure of genomes. Genomes are the complete set of DNA within one cell in an organism.

In addition, graph theory, braid theory and knot theory have proved to be invaluable in studying coiled DNA. Differential geometry has also been used to study the relation between writhe, twist, and linking number.

Finally, network theory has been used to study interactions between genes and proteins; the nodes of the networks in this case are genes or proteins and the edges describe allele combinations that control specific phenotypes.